Question: What is the average rate of change of $h(x)=2^{x+1}$ over the interval $[2,4]$ ?
Answer: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We will need to know the values of $h(2)$ and $h(4)$ to find the slope. $\begin{aligned} h(2)&=2^{(2+1)} \\\\ &=8 \\\\\\ h(4)&=2^{(4+1)} \\\\ &=32 \\\\\\ \dfrac{h(4)-h(2)}{4-2}&=\dfrac{32-8}{2} \\\\ &=12 \end{aligned}$ The average rate of change of $h$ over the interval $[2,4]$ is $12$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${10}$ ${15}$ ${20}$ ${25}$ ${30}$ $y$ $x$ $(2,h(2))$ $(4,h(4))$ secant line